Observe that \(a^{\ast }\)  is the smallest projection \(e\)  such that \(ae = a\) ;
a dual result holds for \(a^{+}\) .
Ehresmann semigroups will be denoted by \((S,U)\)  to make the set of projections clear or by \(S\)  alone if the set of projections
is already evident.
A morphism of Ehresmann semigroups \((S,U)\)  and \((T,V)\)  is a semigroup homomorphism \(\theta \colon S \rightarrow T\) 
such that \(\theta (a)^{\ast } = \theta (a^{\ast })\)  and \(\theta (a)^{+} = \theta (a^{+})\) 
for all \(a \in S\) .
Observe that if \(e \in U\)  then \(\theta (e)^{\ast } = \theta (e^{\ast }) = \theta (e)\) .
It follows that \(\theta \)  maps \(U\)  to \(V\) .
We say that such a morphism is an isomorphism if \(\theta \)  is also an isomorphism of semigroups.
Observe that such an isomorphism induces an isomorphism between \(U\)  and \(V\) .
Ehresmann semigroups can be equivalently described as algebras of type \((2,1,1)\) ;
see [1]}.
The fact that Ehresmann semigroups thereby form a variety means that free objects exist.
We refer the reader to [2]} and [3]} for more information;
we also recommend the references to be found in both these papers.
